Harry’s dragon is last, so he doesn’t witness any of the other champions’s tasks directly. Instead, we learn about how they tackled them from Ron, as she explains it to Harry in the tent, after he finishes the task:
“Cedric did this weird thing where he Transfigured a rock on the ground… turned it into a dog… he was trying to make the dragon go for the dog instead of him. Well, it was a pretty cool bit of Transfiguration, and it sort of worked, because he did get the egg, but he got burned as well – the dragon changed its mind halfway through and decided it would rather have him than the Labrador; he only just got away.
“And that Fleur girl tried this sort of charm, I think she was trying to put it into a trance – well, that kind of worked too, it went all sleepy, but then it snored, and this great jet of flame shot out, and her skirt caught fire – she put it out with a bit of water out of her wand.
“And Krum – you won’t believe this, but he didn’t even think of flying! He was probably the best after you, though. Hit it with some sort of spell right in the eye. Only thing is, it went trampling around in agony and squashed half the real eggs – they took marks off for that, he wasn’t supposed to do any damage to them.”
— Goblet of Fire, chapter 20, The First Task
So Cedric used Transfiguration, Fleur used a sleeping charm and Krum attacked the eyes.
We learn the name of Krum’s spell a bit later, when Sirius writes to Harry, and Hermione can’t help but interject:
Congratulations on getting past the Horntail. Whoever put your name in that goblet shouldn’t be feeling too happy right now! I was going to suggest a Conjunctivitus Curse, as a dragon’s eyes are its weakest point—
“That’s what Krum did!” Hermione whispered—
— Goblet of Fire, chapter 23, The Yule Ball